Abstract: This paper relates the existence of stationary sunspot equilibria in the vicinity of a conventional stationary state to the phase diagram surrounding the latter. We find that the local condition required for a unique equilibrium path returning to the stationary state following any slight disturbance of initial conditions is also sufficient to exclude stationary sunspot equilibria in the stationary state's immediate vicinity. In that sense, an eigenvalue condition familiar from dynamic analysis can tell us something about the possibilities of local sunspot activity as well.